Chapter 13.6, Problem 33WE

To determine

To Prove: The given identity.

Explanation of Solution

Given:

The given identity is ${\cos }^2\theta -{\sin }^2\theta =1-2{\sin }^2\theta$ .

Calculation:

Consider the given identity is,

  ${\cos }^2\theta -{\sin }^2\theta =1-2{\sin }^2\theta$

Solve the above equation as,

  ${\cos }^2\theta -{\sin }^2\theta =1-2{\sin }^2\theta$

By the trigonometric identity the given equation is,

  $\begin{array}{l}1-{\sin }^2\theta -{\sin }^2\theta =1-2{\sin }^2\theta \\ 1-2{\sin }^2\theta =1-2{\sin }^2\theta \end{array}$

Hence, the given identity is proved.